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The following can be stated as a sentence of Morse-Kelley set theory:

If L is the logic of a proper class sized Kripke frame, then L is the logic of a set sized Kripke frame.

It follows from a $\Pi^1_1$ reflection principle (or suitable large cardinal assumption). But I was wondering if this is something that might follow from standard results about frame validity and so be a theorem of MK? And if not, what is the consistency strength of this principle?

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    $\begingroup$ What exactly is "second order $\mathsf{ZFC}$" here? Do you mean some appropriate class theory like $\mathsf{MK}$, or do you really mean the second-order-logic version of $\mathsf{ZFC}$? $\endgroup$
    – Noah Schweber
    Commented Sep 19, 2021 at 19:08
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    $\begingroup$ I meant second order ZFC: an axiomatic system of second order logic with single universally quantified axioms instead of the replacement and separation schemas. I don't think there's much difference mathematically between it and Morse Kelley though, in the sense that you can translate straightforwardly between them. (I'll find a reference for that later.) $\endgroup$
    – Andrew Bacon
    Commented Sep 19, 2021 at 23:57
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    $\begingroup$ @none It's not (necessarily) categorical, it has $V_\kappa$ as standard models for all inaccessible $\kappa$. $\endgroup$
    – Wojowu
    Commented Sep 20, 2021 at 20:16
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    $\begingroup$ I believe this is the Thomason paper @EmilJeřábek mentioned above. $\endgroup$
    – Noah Schweber
    Commented Sep 20, 2021 at 22:55
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    $\begingroup$ @NoahSchweber Wow, I completely forgot about that. I guess we are even, then. $\endgroup$
    – Emil Jeřábek
    Commented Sep 21, 2021 at 15:47

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